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Integration change of variables

  1. Jan 6, 2008 #1
    [SOLVED] Integration change of variables

    1. The problem statement, all variables and given/known data

    An electron is confined in an infinite one dimensional well where 0 < x < L with L = 2 x 10^-10m. Use lowest order perturbation theory to determine the shift in the third level due to the perturbation:

    [tex]V(x) = V_{0}\left(\frac{x}{L}\right)^{2}[/tex]

    where [tex]V_{0} = 0.01eV[/tex].

    After a change of variables, the following integral will be useful:

    [tex]\int^{3\pi}_{0}\phi^{2}sin^{2}\phi d\phi = \frac{9}{2}\pi^{3} - \frac{3}{4}\pi[/tex]


    3. The attempt at a solution

    I've evaluated this question into the following integral:

    [tex]\Delta E_{3}^{(3)} = \frac{2}{L}\frac{V_{0}}{L^{2}}\int^{L}_{0} x^{2} sin^{2}\left(\frac{3\pi x}{L}\right) dx[/tex]

    However I have no idea how to "change variables" with an integral like this, let alone how to get the limits change from 0-L to 0-3pi. Can anyone offer assistance? Many thanks in advance...
     
  2. jcsd
  3. Jan 6, 2008 #2

    nicksauce

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    I believe the change of variables they use is

    [tex]\frac{3\pi x}{L} \rightarrow \phi[/tex]

    So clearly as [tex]x \rightarrow L[/tex] , [tex]\phi \rightarrow 3\pi[/tex]
     
  4. Jan 6, 2008 #3
    hmm, if that was the case, wouldn't the equation take the form

    [tex]\Delta E_{3}^{(3)} = \frac{2}{L}\frac{V_0}{L^2}\frac{L^{3}}{(3\pi)^{3}}\int^{3\pi}_{0}\phi^{2}sin^{2}\phi d\phi[/tex]

    (Note the L^3/... prefactor) ??
     
  5. Jan 6, 2008 #4

    nicksauce

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    So then can't you plug in for
    [tex]\int^{3\pi}_0 \phi^2 \sin^2{\phi}d\phi[/tex]
    which you are given? I don't see the problem.
     
  6. Jan 6, 2008 #5
    ^ I can, I was just seeking assurance that I'd calculated that L^3 prefactor correctly after substitution...
     
  7. Jan 6, 2008 #6

    nicksauce

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    It all looks correct... I know nothing about perturbation theory, so I can't say if the answer qualitatively makes sense with no L dependence.
     
  8. Jan 6, 2008 #7
    Yep, that works good, thanks for your help nicksauce!
     
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